Probability theory and decision theory

This page about probability theory and decision theory will deal about Bayesian, that is, subjective probability, and its use in normative decision analysis.

• Probability theory [frequentist and subjective (Bayesian) approach] Marja-Leenalla sopivia kirjoja? Uncertainty 220-230. Jouni
• Decision theory (utility, decision tree, utilitarian, egalitarian and other decision rules) Decision theory

Two approaches in probability

• Frequentist: Probability is the frequency distribution of outcomes, when a large number of similar events is repeated and obseverd, such tossing a coin, throwing a dice, or taking a blood test from a random sample of individuals in a population.
• Bayesian: When you are indifferent about choosing between A and B, your probability for U1 given A is E.

Basic rules in probability

The probability of A happening = p(A)

p(A)+p(not A)=1

p(A or B) = p(A)+p(B)-p(A and B)

If A and B are mutually exclusive: p(A)+p(B)

p(A|B) = p(A given B) = p(A and B)/p(B)

p(A and B) = p(A|B)p(B)

If A and B not dependent (uncorrelated): p(A)p(B)

The probability of causation (POC)

POC = P(risk from the cause of interest)/P(risk due to all causes)

What is the POC for PM2.5 causing cardiovasular deaths in the EU?

What is the POC for vinylchloride causing angiosarcoma in liver?

POC(PM2.5 causing cardiovasular deaths in the EU):
350 000 deaths/a / 2 000 000 deaths/a = 0.175

"Internationally: Worldwide incidence also is low. Reports show angiosarcoma of the liver with an annual incidence of 2 cases per 10 million population, rising to 2.5 cases per 1000 population in workers exposed to vinyl chloride." 

POC(vinylchloride causing angiosarcoma in liver):
POC = (2.5*10-3 - 2*10-7)/(2.5*10-3) = 0.99992

Bayes' rule

Decision rules:

• Utilitarian: maxd(E(∑(ud,i)))
• Egalitarian: maxd(E(min(ud,i)))
• Elitist: maxd(E(max(ud,i)))

u=utility, d=decision, i=individual

• Cost benetit analysis
• Cost effectiveness analysis